Typo in equation

doc/pub/Regression/html/._Regression-bs090.html

@@ -374,12 +374,9 @@ $$
 \begin{align}
 \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
 &=  \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&= 
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} 
+&=  \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} 
 \tag{22}\\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
 \tag{23}
 \end{align}
 $$

doc/pub/Regression/html/Regression-reveal.html

@@ -3011,19 +3011,18 @@ Using the
 definition of the blocking transformation and the distributive
 property of the covariance, it is clear that since \( h =|i-j| \)
 we can define
+<p>&nbsp;<br>
 $$
 \begin{align}
 \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
 &=  \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&= 
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} 
+&=  \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} 
 \tag{22}\\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
 \tag{23}
 \end{align}
 $$
+<p>&nbsp;<br>
 
 <p>
 The quantity \( \vec{X} \) is asymptotic uncorrelated by assumption, \( \vec{X}_k \) is also asymptotic uncorrelated. Let's turn our attention to the variance of the sample mean \( V(\overline{X}) \).

doc/pub/Regression/html/Regression-solarized.html

@@ -2969,12 +2969,9 @@ $$
 \begin{align}
 \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
 &=  \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&= 
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} 
+&=  \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} 
 \label{_auto12}\\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
 \label{_auto13}
 \end{align}
 $$

doc/pub/Regression/html/Regression.html

@@ -2974,12 +2974,9 @@ $$
 \begin{align}
 \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
 &=  \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&= 
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} 
+&=  \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} 
 \label{_auto12}\\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
 \label{_auto13}
 \end{align}
 $$

doc/pub/Regression/ipynb/Regression.ipynb

@@ -3837,9 +3837,7 @@
     "\n",
     "$$\n",
     "\\begin{equation} \n",
-    "= \n",
-    "\\begin{cases}\n",
-    "\\frac{1}{2}\\gamma_{k}(2h) + \\frac{1}{2}\\gamma_k(2h+1) \\qquad\\qquad\\quad \\ \\ \\text{if $h = 0$} \n",
+    "=  \\frac{1}{2}\\gamma_{k}(2h) + \\frac{1}{2}\\gamma_k(2h+1) \\hspace{0.1cm} \\mathrm{h = 0} \n",
     "\\label{_auto12} \\tag{22}\n",
     "\\end{equation}\n",
     "$$"
@@ -3854,8 +3852,7 @@
     "\n",
     "$$\n",
     "\\begin{equation} \n",
-    "\\frac{1}{4}\\gamma_k(2h-1) + \\frac{1}{2}\\gamma_k(2h) + \\frac{1}{4}\\gamma_k(2h+1) \\quad \\text{else}\n",
-    "\\end{cases}.\n",
+    "=\\frac{1}{4}\\gamma_k(2h-1) + \\frac{1}{2}\\gamma_k(2h) + \\frac{1}{4}\\gamma_k(2h+1) \\quad \\mathrm{else}\n",
     "\\label{_auto13} \\tag{23}\n",
     "\\end{equation}\n",
     "$$"

doc/src/Regression/Regression.do.txt

@@ -2420,11 +2420,8 @@ we can define
 \begin{align}
 \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
 &=  \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&= 
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} \\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=  \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} \\
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
 \end{align}
 !et